Example 1

Let $f(x, y) = k$. Prove that $\lim_{(x, y) \to (a,b)} k = k$.

First note that the function $f(x, y) = k$ represents the plane $z = k$ which is parallel and elevated $k$ units above/below the $xy$-plane. Thus, every point on this plane has $z$-coordinate $k$. Thus, if $(x, y) \to (a,b)$, we can intuitively see that $z = f(x, y) \to k$.